An Approach to Equilibrium

In this paper the concept of statistical equilibrium of an isolated mechanical system chosen from a Gibbsian ensemble of such systems is modified to mean a state in which the observable local macroscopic properties such as the local density of particles, mean energy, temperature, entropy, etc., have attained their equilibrium values, i.e., the values which would be obtained from the density in phase space which corresponds to the appropriate stationary solution of Liouville's equation. The approach to equilibrium in time in this sense (i.e., for the local properties which are suitable averages over the action variables of the system) of a wide class of multiply periodic systems, etc., is demonstrated under the condition that a local property exist initially. Certain characteristics of such a weak convergence as monotonicity of approach after a sufficiently long time has passed and the asymptotic magnitudes of the local properties are investigated. The question of extension of this ergodic result to less restricted mechanical systems has been discussed and the relationship to ergodic theory and coarse graining indicated.On the basis of the above result the general transport theory and the approach to spatial uniformity of a sufficiently ideal gas is investigated, by using as a model a gas of elastic hard-sphere molecules with vanishing diameter. In particular it is shown that under suitable conditions the diffusion in density satisfies Fick's laws and the conduction of heat satisfies Fourier's law, albeit with time-dependent transport coefficients.